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Preliminaries Lines Colouring Quantum Walks

Quantum Problems

Chris Godsil University of Waterloo Plzeň, 4 October, 2016

Chris Godsil University of Waterloo Quantum Problems

Preliminaries Lines Colouring Quantum Walks

Outline

1

Preliminaries

2

Lines Equiangular Lines Mutually unbiased bases

3

Colouring

4

Quantum Walks Basics Which vertices are involved in PST? Near Enough

Chris Godsil University of Waterloo Quantum Problems

Preliminaries Lines Colouring Quantum Walks

Cosmology

Quote Hydrogen is a colorless, odorless gas which given sufficient time, turns into people. (Henry Hiebert)

Chris Godsil University of Waterloo Quantum Problems

Preliminaries Lines Colouring Quantum Walks

Axioms

Quote “The axioms of quantum physics are not as strict as those of mathematics”

Chris Godsil University of Waterloo Quantum Problems

Preliminaries Lines Colouring Quantum Walks Equiangular Lines Mutually unbiased bases

Outline

1

Preliminaries

2

Lines Equiangular Lines Mutually unbiased bases

3

Colouring

4

Quantum Walks Basics Which vertices are involved in PST? Near Enough

Chris Godsil University of Waterloo Quantum Problems

Preliminaries Lines Colouring Quantum Walks Equiangular Lines Mutually unbiased bases

Complex lines

A line in Cd can be represented by any vector that spans it. If x spans a line then P = (x∗x)−1xx∗ represents orthogonal projection onto the line spanned by x (and is independent of the choice of spanning vector).

Chris Godsil University of Waterloo Quantum Problems

Preliminaries Lines Colouring Quantum Walks Equiangular Lines Mutually unbiased bases

Angles between complex lines

Definition The angle between the lines spanned by unit vectors x and y is determined by |x, y| = |x∗y|. If P and Q are the projections xx∗ and yy∗, then tr(PQ) = tr(xx∗yy∗) = tr(y∗xx∗y) = |x, y|2.

Chris Godsil University of Waterloo Quantum Problems

Preliminaries Lines Colouring Quantum Walks Equiangular Lines Mutually unbiased bases

Equiangular sets of lines

Definition A set of complex lines in Cd given by projections P1, . . . , Pn is equiangular if there is a real scalar α such that tr(PrPs) = α whenever r = s. Theorem The maximum size of an equiangular set of lines in Cd is d2. Physicists refer to an equiangular set of d2 lines in Cd as a SIC-POVM.

Chris Godsil University of Waterloo Quantum Problems

Preliminaries Lines Colouring Quantum Walks Equiangular Lines Mutually unbiased bases

Equiangular sets of lines

Definition A set of complex lines in Cd given by projections P1, . . . , Pn is equiangular if there is a real scalar α such that tr(PrPs) = α whenever r = s. Theorem The maximum size of an equiangular set of lines in Cd is d2. Physicists refer to an equiangular set of d2 lines in Cd as a SIC-POVM. [It’s better not to ask.]

Chris Godsil University of Waterloo Quantum Problems

Preliminaries Lines Colouring Quantum Walks Equiangular Lines Mutually unbiased bases

A conjecture

Conjecture For each d ≥ 2, there is a set of d2 equiangular lines in Cd. A more refined version of this asserts that the set of lines is an

• rbit under the action of the Weyl-Heisenberg group.

Chris Godsil University of Waterloo Quantum Problems

Preliminaries Lines Colouring Quantum Walks Equiangular Lines Mutually unbiased bases

A question about chromatic number

Let X(d) be the graph on lines in Cd, where lines given by projections P and Q are adjacent if tr(PQ) = (d + 1)−1. Then ω(X(d)) ≤ d2. Problem What is the chromatic number of X(d)? If there is an equiangular set of d2 lines in Cd then tr(PQ) = (d + 1)−1.

Chris Godsil University of Waterloo Quantum Problems

Preliminaries Lines Colouring Quantum Walks Equiangular Lines Mutually unbiased bases

Outline

1

Preliminaries

2

Lines Equiangular Lines Mutually unbiased bases

3

Colouring

4

Quantum Walks Basics Which vertices are involved in PST? Near Enough

Chris Godsil University of Waterloo Quantum Problems

Preliminaries Lines Colouring Quantum Walks Equiangular Lines Mutually unbiased bases

Flat matrices

We now concern ourselves with orthonormal bases in Cd. Any such basis can be presented as the columns of a d × d unitary matrix. Definition A square (real or complex) matrix is flat if all its entries have the same absolute value.

Chris Godsil University of Waterloo Quantum Problems

Preliminaries Lines Colouring Quantum Walks Equiangular Lines Mutually unbiased bases

Flat matrices

We now concern ourselves with orthonormal bases in Cd. Any such basis can be presented as the columns of a d × d unitary matrix. Definition A square (real or complex) matrix is flat if all its entries have the same absolute value. A flat real orthogonal matrix is a scalar multiple of a Hadamard matrix.

Chris Godsil University of Waterloo Quantum Problems

Preliminaries Lines Colouring Quantum Walks Equiangular Lines Mutually unbiased bases

Flat matrices

We now concern ourselves with orthonormal bases in Cd. Any such basis can be presented as the columns of a d × d unitary matrix. Definition A square (real or complex) matrix is flat if all its entries have the same absolute value. A flat real orthogonal matrix is a scalar multiple of a Hadamard matrix. A flat unitary matrix is a scalar multiple of a complex Hadamard matrix.

Chris Godsil University of Waterloo Quantum Problems

Preliminaries Lines Colouring Quantum Walks Equiangular Lines Mutually unbiased bases

Mutually unbiased bases

Definition Let β1 and β2 be two orthonormal bases in Cd, and let U1 and U2 respectively be unitary matrices whose columns are β1 and β2. The two bases are mutually unbiased if the product U∗

1 U2 is flat.

Equivalently the two bases are mutually unbiased if the change-of-basis matrix is flat. (It is necessarily unitary.)

Chris Godsil University of Waterloo Quantum Problems

Preliminaries Lines Colouring Quantum Walks Equiangular Lines Mutually unbiased bases

Sets of mutually unbiased bases

Theorem The maximum size of a set of mutually unbiased bases in Cd is d + 1.

Chris Godsil University of Waterloo Quantum Problems

Preliminaries Lines Colouring Quantum Walks Equiangular Lines Mutually unbiased bases

Sets of mutually unbiased bases

Theorem The maximum size of a set of mutually unbiased bases in Cd is d + 1. A set of d + 1 mutually unbiased bases in Cd are only known to exist if d is a prime power.

Chris Godsil University of Waterloo Quantum Problems

Preliminaries Lines Colouring Quantum Walks Equiangular Lines Mutually unbiased bases

Sets of mutually unbiased bases

Theorem The maximum size of a set of mutually unbiased bases in Cd is d + 1. A set of d + 1 mutually unbiased bases in Cd are only known to exist if d is a prime power. All known examples can be constructed using commutative semifields and translation planes.

Chris Godsil University of Waterloo Quantum Problems

Preliminaries Lines Colouring Quantum Walks Equiangular Lines Mutually unbiased bases

Sets of mutually unbiased bases

Theorem The maximum size of a set of mutually unbiased bases in Cd is d + 1. A set of d + 1 mutually unbiased bases in Cd are only known to exist if d is a prime power. All known examples can be constructed using commutative semifields and translation planes. If d is not a prime power then, in most cases, the best lower bound is three.

Chris Godsil University of Waterloo Quantum Problems

Preliminaries Lines Colouring Quantum Walks Equiangular Lines Mutually unbiased bases

Cayley graphs

Definition Suppose G is a group and C ⊆ G such that 1 / ∈ C and g−1 ∈ C for each g in C. The Cayley graph X(G, C) with connection set C is the graph with vertex set G, where group elements g and h are adjacent if hg−1 ∈ C.

Chris Godsil University of Waterloo Quantum Problems

Preliminaries Lines Colouring Quantum Walks Equiangular Lines Mutually unbiased bases

MUBs and cliques

Let F denote the set of flat unitary matrices of order d × d and let U(d) be the unitary group. Then the maximum size of a set of mutually unbiased bases in Cd is the maximum size of a clique in the Cayley graph X(U(d), F).

Chris Godsil University of Waterloo Quantum Problems

Preliminaries Lines Colouring Quantum Walks

A finite subgraph of the unit sphere in Rd

Definition Let Φ(d) denote the graph with the ±1-vectors of length d as its vertices, where two vectors are adjacent if and only if they are

• rthogonal.

Chris Godsil University of Waterloo Quantum Problems

Preliminaries Lines Colouring Quantum Walks

A finite subgraph of the unit sphere in Rd

Definition Let Φ(d) denote the graph with the ±1-vectors of length d as its vertices, where two vectors are adjacent if and only if they are

• rthogonal.

1 If d is odd, Φ(d) has no edges. Chris Godsil University of Waterloo Quantum Problems

Preliminaries Lines Colouring Quantum Walks

A finite subgraph of the unit sphere in Rd

Definition Let Φ(d) denote the graph with the ±1-vectors of length d as its vertices, where two vectors are adjacent if and only if they are

• rthogonal.

1 If d is odd, Φ(d) has no edges. 2 If d ≡ 2 modulo four, Φ(d) is bipartite. Chris Godsil University of Waterloo Quantum Problems

Preliminaries Lines Colouring Quantum Walks

A finite subgraph of the unit sphere in Rd

Definition Let Φ(d) denote the graph with the ±1-vectors of length d as its vertices, where two vectors are adjacent if and only if they are

• rthogonal.

1 If d is odd, Φ(d) has no edges. 2 If d ≡ 2 modulo four, Φ(d) is bipartite. 3 α(Φ(d)) ≤ 2d

d and thus χ(Φ(d)) ≥ d; hence if χ(Φ(d)) = d,

then d is a power of two.

Chris Godsil University of Waterloo Quantum Problems

Preliminaries Lines Colouring Quantum Walks

A finite subgraph of the unit sphere in Rd

Definition Let Φ(d) denote the graph with the ±1-vectors of length d as its vertices, where two vectors are adjacent if and only if they are

• rthogonal.

1 If d is odd, Φ(d) has no edges. 2 If d ≡ 2 modulo four, Φ(d) is bipartite. 3 α(Φ(d)) ≤ 2d

d and thus χ(Φ(d)) ≥ d; hence if χ(Φ(d)) = d,

then d is a power of two.

4 ω(Φ(d)) ≤ d, equality holds if and only if a Hadamard matrix

exists.

Chris Godsil University of Waterloo Quantum Problems

Preliminaries Lines Colouring Quantum Walks

The chromatic number of Φ(d) increases exponentially

Theorem (Frankl and Rödl) There is a constant c such that 0 < c < 2 and if 4|d and d is large enough, then α(Φ(d)) < cd.

Chris Godsil University of Waterloo Quantum Problems

Preliminaries Lines Colouring Quantum Walks

The chromatic number of Φ(d) increases exponentially

Theorem (Frankl and Rödl) There is a constant c such that 0 < c < 2 and if 4|d and d is large enough, then α(Φ(d)) < cd. So χ(Φ(d)) is exponential in d.

Chris Godsil University of Waterloo Quantum Problems

Preliminaries Lines Colouring Quantum Walks

Homomorphisms of graphs

Definition If X and Y are graphs, a homomorphism from X to Y is a map ϕ from V (X) to V (Y ) such that if a and b are adjacent vertices in X, then ϕ(a) and ϕ(b) are adjacent in Y . A homomorphism from X into the complete graph Kd is the same thing as a proper d-coloring of X.

Chris Godsil University of Waterloo Quantum Problems

Preliminaries Lines Colouring Quantum Walks

Quantum colorings

Definition Let D be the subset of U(d) consisting of the unitary matrices with zero diagonal. A graph X has a quantum d-coloring if it admits a homomorphism in the Cayley graph X(U(d), D). Theorem The graph Φ(d) admits a quantum d-coloring.

Chris Godsil University of Waterloo Quantum Problems

Preliminaries Lines Colouring Quantum Walks Basics Which vertices are involved in PST? Near Enough

Outline

1

Preliminaries

2

Lines Equiangular Lines Mutually unbiased bases

3

Colouring

4

Quantum Walks Basics Which vertices are involved in PST? Near Enough

Chris Godsil University of Waterloo Quantum Problems

Preliminaries Lines Colouring Quantum Walks Basics Which vertices are involved in PST? Near Enough

Classical discrete walks

Let X be a graph with adjacency matrix A and let ∆ be the diagonal matrix with i-th diagonal entry equal to the valency of i-th vertex of X. Assume there are no vertices of valency zero. If we set M = ∆−1A, then M determines a discrete random walk

• n the vertices of X. The i-th row of Mk is a probability density
• n V (X); thus for each vertex there is a sequence of densities,

indexed by the non-negative integers.

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Classical continuous walks

A continuous random walk on X is determined by the matrices exp(−t(∆ − A)), for non-negative real values of t. Again at a fixed time t, each row of this matrix is a probability density on V (X).

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Preliminaries Lines Colouring Quantum Walks Basics Which vertices are involved in PST? Near Enough

Discrete quantum walks

We use M ◦ N to denote the Schur product of two matrices of the same order. If U is a unitary matrix, then the matrix U ◦ U is non-negative and each row and column sums to 1. So the matrices Uk ◦ Uk (for k = 0, 1, . . .) provide us with sequences of probability densities

• n the columns of U, one for each row. It is not unreasonable to

view these sequences as discrete quantum walks. (In practice we need to be able to express U as a product of sparse unitary matrices.)

Chris Godsil University of Waterloo Quantum Problems

Preliminaries Lines Colouring Quantum Walks Basics Which vertices are involved in PST? Near Enough

Continuous quantum walks

Let X be a graph with adjacency matrix A. We define matrices U(t) by U(t) = exp(itA). (As usual the exponential is defined by a power series.) We see that U(t)∗ = exp(−itA) = U(t)−1 and so U(t) is unitary for all t. We say that U(t) determines a continuous quantum walk on X.

Chris Godsil University of Waterloo Quantum Problems

Preliminaries Lines Colouring Quantum Walks Basics Which vertices are involved in PST? Near Enough

Example: K2

Since A(K2) =

• 1

1

• ,

we have Ak = I if k is even and Ak = A if k is odd. Hence exp(itA) = cos(t)

• 1

1

• + i sin(t)
• 1

1

• .

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Preliminaries Lines Colouring Quantum Walks Basics Which vertices are involved in PST? Near Enough

Special times for K2

uniform mixing UK2(π/4) = 1 √ 2

• 1

1 1 1

• .

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Preliminaries Lines Colouring Quantum Walks Basics Which vertices are involved in PST? Near Enough

Special times for K2

uniform mixing UK2(π/4) = 1 √ 2

• 1

1 1 1

• .

perfect state transfer UK2(π/2) =

• i

i

• .

Chris Godsil University of Waterloo Quantum Problems

Preliminaries Lines Colouring Quantum Walks Basics Which vertices are involved in PST? Near Enough

Special times for K2

uniform mixing UK2(π/4) = 1 √ 2

• 1

1 1 1

• .

perfect state transfer UK2(π/2) =

• i

i

• .

periodicity UK2(π) =

• −1

−1

• .

Chris Godsil University of Waterloo Quantum Problems

Preliminaries Lines Colouring Quantum Walks Basics Which vertices are involved in PST? Near Enough

Mixing: Observations and Question

For which graphs X is there a vertex a and a time t such that U(t)ea is flat?

Chris Godsil University of Waterloo Quantum Problems

Preliminaries Lines Colouring Quantum Walks Basics Which vertices are involved in PST? Near Enough

Mixing: Observations and Question

For which graphs X is there a vertex a and a time t such that U(t)ea is flat? For which graphs X is there a time t such that U(t)ea is flat?

Chris Godsil University of Waterloo Quantum Problems

Preliminaries Lines Colouring Quantum Walks Basics Which vertices are involved in PST? Near Enough

Mixing: Observations and Question

For which graphs X is there a vertex a and a time t such that U(t)ea is flat? For which graphs X is there a time t such that U(t)ea is flat? We have uniform mixing on Kn if and only if n ∈ {2, 3, 4}.

Chris Godsil University of Waterloo Quantum Problems

Preliminaries Lines Colouring Quantum Walks Basics Which vertices are involved in PST? Near Enough

Mixing: Observations and Question

For which graphs X is there a vertex a and a time t such that U(t)ea is flat? For which graphs X is there a time t such that U(t)ea is flat? We have uniform mixing on Kn if and only if n ∈ {2, 3, 4}. The only even cycle that admits uniform mixing is C4.

Chris Godsil University of Waterloo Quantum Problems

Preliminaries Lines Colouring Quantum Walks Basics Which vertices are involved in PST? Near Enough

Mixing: Observations and Question

For which graphs X is there a vertex a and a time t such that U(t)ea is flat? For which graphs X is there a time t such that U(t)ea is flat? We have uniform mixing on Kn if and only if n ∈ {2, 3, 4}. The only even cycle that admits uniform mixing is C4. Conjecture (N. Mullin): If d ≥ 5, no Cayley graph for Zn

d

admits uniform mixing.

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Perfect state transfer

Definition We have perfect state transfer from vertex a to vertex b at time t if |U(t)a,b| = 1. So we have perfect state transfer between the two vertices of K2 at time π/2.

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Composite Systems

If X and Y are graphs and we run walks on them independently; the composite quantum system is described by UX(t) ⊗ UY (t).

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Preliminaries Lines Colouring Quantum Walks Basics Which vertices are involved in PST? Near Enough

Composite Systems

If X and Y are graphs and we run walks on them independently; the composite quantum system is described by UX(t) ⊗ UY (t). The Cartesian product of X and Y has adjacency matrix AX ⊗ I + I ⊗ AY . Since AX ⊗ I and I ⊗ AY commute, UXY (t) = UX(t) ⊗ UY (t).

Chris Godsil University of Waterloo Quantum Problems

Preliminaries Lines Colouring Quantum Walks Basics Which vertices are involved in PST? Near Enough

An Example: Cartesian Product

Figure: P4 P4

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Preliminaries Lines Colouring Quantum Walks Basics Which vertices are involved in PST? Near Enough

More state transfer

Theorem If X and Y admit perfect state transfer at time τ, then their Cartesian product X Y admits perfect state transfer at time τ. Corollary The d-cube admits perfect state transfer between antipodal vertices at time π/2.

Chris Godsil University of Waterloo Quantum Problems

Preliminaries Lines Colouring Quantum Walks Basics Which vertices are involved in PST? Near Enough

Outline

1

Preliminaries

2

Lines Equiangular Lines Mutually unbiased bases

3

Colouring

4

Quantum Walks Basics Which vertices are involved in PST? Near Enough

Chris Godsil University of Waterloo Quantum Problems

Preliminaries Lines Colouring Quantum Walks Basics Which vertices are involved in PST? Near Enough

Another view of perfect state transfer

If a ∈ V (X) we use ea to denote the characteristic vector of a, viewed as a subset of V (X). So the vectors ea for a in V (X) form the standard basis for CV (X). We have pst from a to b at time t if there is a complex scalar γ (necessarily of norm 1) such that U(t)ea = γeb.

Chris Godsil University of Waterloo Quantum Problems

Preliminaries Lines Colouring Quantum Walks Basics Which vertices are involved in PST? Near Enough

Counting walks

Since U(t) and A commute we have γAkeb = AkU(t)ea = U(t)Akea and since U(t) preserves length, eb, Akeb = ea, Akea. Lemma If we have perfect state transfer from vertex a to vertex b in X (for some t), then for all non-negative integers k the number of closed walks in X of length k that start at a is equal to the number of closed walks in X of length k that start at b.

Chris Godsil University of Waterloo Quantum Problems

Preliminaries Lines Colouring Quantum Walks Basics Which vertices are involved in PST? Near Enough

Counting walks

Since U(t) and A commute we have γAkeb = AkU(t)ea = U(t)Akea and since U(t) preserves length, eb, Akeb = ea, Akea. Lemma If we have perfect state transfer from vertex a to vertex b in X (for some t), then for all non-negative integers k the number of closed walks in X of length k that start at a is equal to the number of closed walks in X of length k that start at b. This implies that X \a and X \b have the same characteristic polynomials.

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Another necessary condition for PST

Theorem If we have perfect state transfer from a to b in X, then an automorphism of X fixes a if and only if it fixes b. Proof. The automorphism P fixes a vertex u if and only if Peu = eu. Suppose U(t)ea = γeb and P fixes a. γPeb = PU(t)ea = U(t)Pea = U(t)ea = γeb. Hence P fixes b.

Chris Godsil University of Waterloo Quantum Problems

Preliminaries Lines Colouring Quantum Walks Basics Which vertices are involved in PST? Near Enough

Outline

1

Preliminaries

2

Lines Equiangular Lines Mutually unbiased bases

3

Colouring

4

Quantum Walks Basics Which vertices are involved in PST? Near Enough

Chris Godsil University of Waterloo Quantum Problems

Preliminaries Lines Colouring Quantum Walks Basics Which vertices are involved in PST? Near Enough

The Geometric View

The set {U(t) : t ∈ R} is a subgroup of the unitary group and the set Ωa = {U(t)eaeT

a U(−t) : t ∈ R}

can be viewed as a curve in complex projective space. We have perfect state transfer from a to b if the curves Ωa and Ωb have a point in common.

Chris Godsil University of Waterloo Quantum Problems

Preliminaries Lines Colouring Quantum Walks Basics Which vertices are involved in PST? Near Enough

The Geometric View

The set {U(t) : t ∈ R} is a subgroup of the unitary group and the set Ωa = {U(t)eaeT

a U(−t) : t ∈ R}

can be viewed as a curve in complex projective space. We have perfect state transfer from a to b if the curves Ωa and Ωb have a point in common. Question: What if these curves are close?

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Preliminaries Lines Colouring Quantum Walks Basics Which vertices are involved in PST? Near Enough

Pretty Good State Transfer

Definition We have pretty good state transfer from a to b if the closures of the orbits Ωa and Ωb have a point in common. Equivalently we have pretty good state transfer from a to b if, given ǫ > 0, there is a time t and a complex scalar γ such that U(t)ea − γeb < ǫ.

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Preliminaries Lines Colouring Quantum Walks Basics Which vertices are involved in PST? Near Enough

PGST on Paths

Theorem The only paths that admit perfect state transfer are P2 and P3.

Chris Godsil University of Waterloo Quantum Problems

Preliminaries Lines Colouring Quantum Walks Basics Which vertices are involved in PST? Near Enough

PGST on Paths

Theorem The only paths that admit perfect state transfer are P2 and P3. Theorem (Godsil, Kirkland, Smith, Severini) The path Pn on n vertices admits pretty good state transfer if and

• nly if n + 1 is a prime, twice a prime, or a power of two.

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Near Enough Implies Equal Stabilizers

Theorem If there is a complex number γ of norm 1 such that U(t)ea − γeb < 1 √ 2 then any automorphism of X that fixes a must fix b.

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Near Enough Implies Cospectral Vertices

Theorem Let n = V (X) and let ρ be the largest eigenvalue of A. If there is a complex number γ of norm 1 and time t such that U(t)ea − γeb < 1 8n2ρ4n , then a and b are cospectral.

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Preliminaries Lines Colouring Quantum Walks Basics Which vertices are involved in PST? Near Enough

Walk Matrices

Definition If X is a graph on n vertices and a ∈ V (X), then the walk matrix Wa is the n × n matrix with the vectors A0ea, . . . , An−1ea as its columns. Lemma The number of closed walks of length k at the vertex a and b are equal for all k if and only if W T

a Wa = WbW T b .

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A Sketch of a Proof

Suppose U(t)ea ≈ γeb. Then U(t)Akea ≈ γAkeb and therefore W T

a Wa ≈ W T b Wb. As these last two matrices have

integer entries, it follows that if U(t)ea is close enough to γeb, then W T

a Wa = W T b Wb and hence a and b are cospectral.

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The Question Is:

What interesting properties of a graph X can we determine from the geometry of its vertex orbits?

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The End(s)

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